A084109 n is congruent to 1 (mod 4) and is not the sum of two squares.

21, 33, 57, 69, 77, 93, 105, 129, 133, 141, 161, 165, 177, 189, 201, 209, 213, 217, 237, 249, 253, 273, 285, 297, 301, 309, 321, 329, 341, 345, 357, 381, 385, 393, 413, 417, 429, 437, 453, 465, 469, 473, 489, 497

Offset

1,1

Comments

Alternatively, n is congruent to 1 (mod 4) with at least 2 distinct prime factors congruent to 3 (mod 4) in the squarefree part of n. - Comment corrected by Jean-Christophe Hervé, Oct 25 2015

Applications to the theory of optimal weighing designs and maximal determinants: An (n+1) X (n+1) conference matrix is impossible.

The upper bound of Ehlich/Wojtas on the determinant of a (0,1) matrix of order congruent to 1 (mod 4) cannot be achieved for n X n matrices.

The bound of Ehlich/Wojtas on the determinant of a (-1,1) matrix of order congruent to 2 (mod 4) cannot be achieved for (n+1) X (n+1) matrices.

Numbers with only odd prime factors, of which a strictly positive even number are raised to an odd power and congruent to 3 (mod 4). - Jean-Christophe Hervé, Oct 24 2015

References

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

Links

Jean-Christophe Hervé, Table of n, a(n) for n = 1..1000

H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964) 123-132.

D. Raghavarao, Some aspects of weighing designs, Ann. Math. Stat. 31 (1960) 878-884.

Example

a(1) = 3*7 = 21, a(2) = 3*11 = 33, a(3) = 3*19 = 57, a(14) = 3^3*7 = 189.

Maple

N:= 1000: # to get all entries <= N
S:= {seq(i, i=1..N, 4)} minus
   {seq(seq(i^2+j^2, j=1..floor(sqrt(N-i^2)), 2), i=0..floor(sqrt(N)), 2)}:
sort(convert(S, list)); # Robert Israel, Oct 25 2015

Mathematica

a[m_] := Complement[Range[1, m, 4], Union[Flatten[Table[j^2+k^2, {j, 1, Sqrt[m], 2}, {k, 0, Sqrt[m], 2}]]]]

Prog

(PARI) is(n)=if(n%4!=1, return(0)); my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(1))); 0 \\ Charles R Greathouse IV, Jul 01 2016

Crossrefs

Cf. A000952, A003432, A003433, A022544, A001481.

Sequence in context: A189986 A190299 A280262 * A016105 A187073 A271101

Adjacent sequences: A084106 A084107 A084108 * A084110 A084111 A084112

Keyword

easy,nonn

Author

William P. Orrick, Jun 18 2003

Status

approved